Regression transients modeling of solid rocket motor burning surfaces with physics-guided neural network

Monitoring the burning surface regression in ground static ignition tests is crucial for predicting the internal ballistic performance of solid rocket motors (SRMs). A previously proposed ultra-sparse computed tomography imaging method provides a possibility for real-time monitoring. However, sample shortages of SRMs highlights the need for monitoring accuracy, especially given the high cost associated with the design and development of SRM systems. Therefore, constructing datasets via regression simulations to compensate for SRM sample shortages is critical. To address this issue, we recommend adopting the level-set method to dynamically track the burning surface by solving partial differential equations (PDEs). The computational cost of numerical solution is prohibitive for scientific applications involving large-scale spatiotemporal domains. The physics-informed neural network (PINN) and neural operator have been used to accelerate the solution of PDE, showing satisfactory prediction performance and high computational efficiency. We designed a physics-guided network, named LS-PhyNet, that couples the potential physical mechanisms of burning surface regression into the deep learning framework. The proposed method is capable of encoding well-established traditional numerical discretization methods into the network architecture to leverage prior knowledge of underlying physics, thus providing the model with enhanced expressive power and interpretability. Experimental results prove that LS-PhyNet can better reproduce the burning surfaces obtained by numerical solution with only small data regimes, providing a new paradigm for real-time monitoring of burning surface regression transients during static ignition tests.


Introduction
With the advantages of fast launch response, strong maneuverability, convenient maintenance and usage, long life, and easy storage, solid rocket motors (SRMs) have proven to be reliable and cost-effective propulsion systems, and they are preferred for current strategic and tactical missile weapon systems as well as large space mission rockets [1].Monitoring the transients of burning surfaces using static ignition tests is a prerequisite for achieving successful flight.Existing technical methods, such as ultrasonic technology [2], have not yet reached to observe real-time dynamic changes in grain burning structures under thermal conditions.
In the previous work, an ultra-sparse computed tomography (CT) imaging method was proposed, which provides the possibility of dynamic monitoring of particle regression in static ignition tests via uninterrupted data acquisition and reconstruction by X-ray2CTNet [3].X-ray2CTNet learns to map 2D projections (obtained from different regression models) to 3D volumes under different burning states using a data-driven strategy.In other words, the idea behind X-ray2CTNet is to utilize regression-simulated data for training and actual grains for testing to achieve fast reconstruction of ultra-sparse CT imaging on the burning surface.The excellent generalization ability of the X-ray2CTNet is based on the assumption that the data distribution of the training set is consistent with that of the test set.Therefore, constructing datasets via regression simulations to compensate for SRM sample shortages is an urgent issue.
Research on the simulation of burning surface regression primarily includes the analytical [4], mesh [5], solid construction [6], minimum distance [7], and level-set (LS) [8,9] methods.Because the burning rate is both spatially and temporally dependent [7], the precise prediction of the SRM performance implies that the regression simulation algorithm is applicable under both uniform and non-uniform burning conditions.Considering the complexity of the burning rate distribution, adapting the analytical, mesh, solid construction, and minimum distance methods to the needs of the burning surface regression calculation is difficult in non-uniform burning rates.In contrast, the LS method is well adapted to the calculation of burning surface regression at non-uniform burning rates, which regards the zero-LS of the signed distance function (SDF) as the active interface.Furthermore, it assesses the evolution of the zero-LS by solving the initial value partial differential equation (PDE) to achieve a dynamic tracking of the SRM burning surface, which is theoretically applicable to the burning surface regression of grains with arbitrary shapes [9].
The initial value PDE for the evolution of the LS function is similar to the Hamilton-Jacobi equation [10].Owing to the inherent effect of the numerical solution, preventing deviations between the LS and SDF of the interface during the calculation process is difficult.Hence, frequent re-initialization operations [11] are required, resulting in high time and computational costs.The computational cost of numerical solution is prohibitive for scientific applications involving large-scale spatiotemporal domains.Therefore, developing a method to accelerate the PDE solution is critical for satisfying computational efficiency requirements, such as frequent modification and structural optimization in engineering practice.
Researchers have attempted to identify new methods for solving PDEs faster, one of which is the neural-network-based solution method.Automatic differentiation [12] can use the chain rule to calculate derivatives precisely to replace complex gradient calculations in PDEs, laying the foundation for solving PDEs based on neural networks.
Neural-network-based PDEs include data-and physics-model-driven methods.For data-driven methods [13][14][15], the exact solution of the PDE must be obtained in advance to ensure the model that can characterize the PDE.Purely data-driven methods directly use neural networks to learn physical laws from the data, leading to problems such as high training costs, error accumulation, and poor generalization.In engineering practice, the available data from in situ monitoring are often sparse, which limits the application of purely data-driven models.The physics-informed neural network (PINN) [16] is designed to handle problems with little or no training data and has shown considerable potential for application in various engineering fields, including fluid mechanics [17], solid mechanics [18], electromagnetic analysis [19], power systems [20], chemistry [21], and others [22][23][24].PINN utilizes prior knowledge of physical laws in the form of a PDE and boundary conditions to construct residuals and embed them into a loss function to generate a neural network model with physical constraints.Although the PINN is a simple and flexible framework for solving the PDE problem, it has many limitations and room for improvement [25].The variants of PINN focus on developing better optimization goals to improve its performance, such as the adaptive activation function [26] and loss function optimization [27].The PINN and its variants mostly adopt the multi-layer perceptron models, which is difficult or inefficient for solving complex physical systems.Thus, on one hand, PINN variants attempt to incorporate long short-term memory (LSTM) [28], convolutional neural network (CNN) [29], autoencoder [30] architectures into the PINN framework.On the other hand, other learning paradigms such as transfer learning [31], meta-learning [32], and multi-task learning [33] are proposed to enhance the performance of PINN.In summary, PINN belongs to a neural solver that constructs a neural network via physical laws to approximate the solution of physical system.This practice can reduce the dependence on or eliminate label data, significantly improving the generalization ability and enabling better interpretability.Recent advances show that neural operator learning can naturally achieve the goal of PDEs solving, e.g.DeepONet [14] and Fourier neural operator [34].With the help of the generalizability of neural networks, neural operators seek for the nonlinear mapping from parametric networks to the numerical solution, not just the solution to a certain instance of the PDEs (e.g.PINNs).However, it is usually more demanding on training data, especially for some complex PDEs.A straightforward way to overcome this drawback is to incorporate the idea of PINNs into the loss function to reduce the data requirements [35,36].
In this work, we propose a network for SRM burning surface regression, named LS-PhyNet.Replacing the traditional numerical method directly with a PINN is not feasible, and combining PINN with traditional solvers may be a fruitful research topic.On the one hand, the loss function that integrates errors from small-sample data, boundary conditions, LS optimization equation, and PDE is defined to implement X Sun et al penalties.On the other hand, a dense encoding-decoding (ED) network is used to recover or approximate the evolution operator of the PDE, which characterizes the time evolution of the solutions.In this study, the PDE is used as implicit prior information, and neural network training is regularized by minimizing the composite loss function.In the case of limited data, the training program was normalized to achieve the regression of the SRM burning surface.
In summary, we make the following contributions: (1) The proposed method encodes the physical information contained in a PDE into a loss function to compensate for the lack of label data, enabling the regression calculation of the SRM burning surface with any time length, uniform burning rate, or non-uniform burning rate.To the best of our knowledge, this is the first study of SRM burning surface regression calculation based on PINN.It enables the optimization of a technique applied to rapid monitoring of SRM burning surface transients to assess the accuracy and performance of SRM, which are critical to several scientific and engineering applications.(2) A novel optimization objective is introduced.The loss function not only punishes prediction errors related to a handful of available data but also enforces the PDE, boundary conditions, and without re-initialization to achieve seamless integration of measurement data and physical constraints.
(3) The model employs ED and dense connections to approximate the evolution operator of the physical system, which characterizes the time evolution of the solutions, rather than directly representing the physical system.The proposed method is capable of encoding well-established traditional numerical discretization methods into the network architecture, thus providing the model with enhanced expressive power and predictive accuracy.
The rest of paper is organized as follows.In section 2, we explain our proposed LS-PhyNet in detail.In section 3, the dataset, training details, and loss function are introduced.In section 4, the results of the experiments are demonstrated.Then the last section provides the discussion and conclusion.

Preliminaries 2.1.1. LS method
The LS method was first developed by Osher and Sethian [10], and it significantly impacts the calculation of interface motion.The interface tracking method based on LS is suitable for burning surface changes in complex grain configurations.Several studies have demonstrated the effectiveness of the LS method for SRM burning surface regression.
The LS method considers the interface of two media moving with time in region Ω as the zero-LS of the implicit function φ (X, t) that satisfies a certain equation.At each time t, obtaining the value of function φ (X, t) is sufficient to determine the position of its zero-LS, that is, the position of the moving interface.The construction of φ (X, t) must satisfy that at any time, the moving interface Γ (t) is the zero-LS of φ (X, t), which means it satisfies (1) The initial value of φ was defined as the SDF from each point in the computational area to the interface.The initial value in the solid region Ω solid is positive, whereas the initial value in the gas region Ω solid is negative (figure 1).φ (X, 0) is the signed distance from point X to interface Γ(0), and it is defined as follows (2): To ensure that the zero-LS of function φ is the motion interface at any time, φ must satisfy the control equation defined in (3) F is the velocity of the moving interface in the normal direction, which is the burning rate of the SRM, and (3) is the governing equation of the initial value LS method.Once the function φ is solved, the points with a distance of 0 are the points on the moving interface.The initial value governing the equation enables the interface velocity F to be positive or negative, which means that the interface can shrink or expand.

LS numerical solution
The governing equation ( 3) is a Hamilton-Jacobi equation [10].The viscosity calculation scheme satisfying the entropy condition of this type of governing equation is given by ( 4).
Here D + and D − denote the central difference operators.
, and the other definitions are similar.h denotes the spatial step.∆t represents the time step, which satisfies the Courant-Friedrichs-Lewy condition to ensure the stability and convergence of the numerical calculation.

Re-initialization
Maintaining the SDF in numerical calculations is crucial.However, whether in a uniform or non-uniform velocity field, φ (X, t) no longer satisfies the accurate signed distance after several time steps and becomes increasingly distorted.To ensure that the function φ (X, t) remains as the SDF to the interface and eliminate the deviation caused by calculation, a method called re-initialization is required to solve the Hamilton-Jacobi equation to a stable state.The re-initialization process uses a new LS function φ (X, t) to replace the current value of φ (X, t) and serves as the initial value for continued iterative calculations.We adopted the method proposed in [37] for re-initialization, as shown in ( 6) The solution of ( 6) is the optimal signed distance to the interface.Here, d 0 represents the current φ (X, t), d represents the φ (X, t) to be solved.
and δ represents the spatial step size.The re-initialization function ( 6) can be solved using the Engquist-Osher discretization scheme [10] as follows: where a + = max (a, 0) , a − = min (a, 0), and the rest have similar definitions.

Proposed physics-guided network
The traditional numerical discretization method for LS involves numerous differential operator calculations and frequent re-initialization steps, resulting in a high computational cost.The proposed method expands (4) into a multilayer deep neural network architecture, namely LS-PhyNet, with the number of layers depending on the iteration, as shown in figure 2. LS-PhyNet is devised with an input layer composed of the initial φ 0 , burning rate F, and time-step ∆t.The output φ t for each layer corresponds to the SRM burning state at a certain time.We consider the differential operators ∇ + (φ ) and ∇ − (φ ) as evolution operators and approximate their operations using a dense convolutional ED network.Importantly, ∆t max F ijk , 0 ∇ + + min F ijk , 0 ∇ − is embedded in the network that participates in forward and backward propagation.The network is designed to preserve a given physical structure, such as governing PDEs and boundary conditions.The prior physics knowledge is forcibly 'encoded' to ensure interpretability of the network.Therefore, the traditional numerical method is replaced by the LS-PhyNet, which approximates the PDE solution.Once the LS-PhyNet is successfully trained, the network predicts in accordance with the time-dependent LS governing equation and satisfies the specified  boundary conditions, making predicting the SRM burning surface regression for any length of time (i.e.any iterations) possible, not limited to a specific time length.
Notably, the dense convolutional ED network is repeated (N − 1) times for (N − 1) iterations before the output φ N is generated, and each network contains the same parameter set.Details on the implementation of dense block, ED block, ED fusion block will be discussed in the following.

Dense block
A dense connection has significant advantages in feature extraction since each channel characteristic of its output is directly associated with the intermediate feature maps, thereby preserving the detailed information of each feature map.Therefore, dense blocks are introduced to create dense connections between the convolution layers, enhancing feature representation.As shown in figure 3, the input and output feature map of the dense block remains the same size, whereas the number of channels increases with depth.A 1 × 1 × 1 convolution layer is used to control the number of output channels self-adaptively.

ED block
The ED block is used to change the size of the feature map while maintaining the same number of feature maps.In the encoding part, information at different levels of the feature maps is encoded to obtain spatiotemporal feature maps of different sizes.These feature maps will be transmitted along different paths to the decoder.In the encoding part, the resulting different scale spatiotemporal feature maps are obtained and combined in the encoding part via the ED fusion module.
As shown in figure 4, the encoding block halves the feature map size to make it a lower-dimensional representation, with the aim to extract as many low-and high-level features as possible.The encoding block doubles the feature map size and fuses the features extracted during the encoding process to complete the feature reconstruction while minimizing information loss.

ED fusion block
To take full advantage of global and local features, some existing ED networks bridge the encoder and decoder by skip connections.Considering that low-level feature maps with low semantics contain many irrelevant and ambiguous details, direct fusion with high-level feature maps via skip connections may interfere with the prediction.The semantic gap is a key challenge in feature fusion.We propose using the ED fusion module to promote the integration of high-level semantic features and low-level fine-grained features via the attention mechanism and discard irrelevant features, which is conducive to achieving a more sophisticated deep feature fusion effect.
As shown in figure 5, low-level features are used to calculate the spatial attention, while high-level features are used to calculate the channel attention.Channel attention is more relevant to a task and requires higher-level features that contain more semantic information.Therefore, channel attention based on high-level semantic information is used to assist in generating spatial attention, which also aids in selecting detailed spatial information to some extent and further guides the fusion of high-and low-level feature maps.

Dataset
The SRMs of four grain propellants (as shown in figure 6) were selected as the initial burning surface, with burning rates F ⊆ {0.5, 1.0, 1.5, 2.0, 2.5, 3.0} and time-step ∆t ⊆ {0.05, 0.1}.By random combination, ∆t ⊆ {0.05, 0.1} groups were generated.The number of iterations was set to 300.Therefore, there were 300 combustion surface regression states for each SRM grain under different initial conditions.Of the 300 burning surface states, 20 were randomly selected to generate 960 label data sets, which were generated by the LS traditional algorithm to construct a dataset under uniform burning rates.In addition, the non-uniform burning rates were set to{0.5, 1.0}, {1.0, 1.5}, {0.8, 1.6}, and{1.5, 2.5}, and a total of 4 × 4 × 2 = 32 sets of data were generated.The same method was used to select 640 sets of label data under non-uniform burning rates.The training and validation sets were randomly selected according to a ratio of 8:2.
In the testing phase, the burning rates, time steps, time range (number of iterations), and initial burning surfaces differ from what are used in the training phase.This would offer more convincing results and support the generalization of the proposed LS-PhyNet.

Training details
In this study, to solve the optimization problem, Adam was used for training optimization.The batch size was set to six, and the initial learning rate was set to 10 −4 .As the number of epochs increased, the learning rate decreased.The sizes of the inputs and outputs are 256 × 256 × 256.All implementations in this study were coded in TensorFlow and performed on an NVIDIA Tesla V100 GPU card (32 G), and 64 GB of RAM using an Ubuntu 16.04 system.The parametric structures of all layers in the LS-PhyNet are shown in table 1.The output size is denoted as N 3 × k, where N is the resolution of feature maps and k is the number of feature map channels.

Physics-constrained loss
For the points located in the computational area Ω, the governing equation of the LS is considered the physical constraint loss residual to minimize the violation of the solution φ to the PDE.This corresponds to a constrained optimization as follows:

Boundary condition loss
As shown in figure 1, in area Ω outside the SRM grain, the φ is 0, that is, B(t) = X ∈ Ω : φ (X, t) = 0 .To make the network model more reasonable, the following constraint is designed

Data loss
The PDE solutions of the SRM grains at different burning rates and times are considered small sample labels.Mean squared error is selected as the data loss function, and corresponded to the following.Here, φ truth represents the true solution and φ pred represents the predicted solution

LS optimization loss
For an accurate SDF, |∇φ | ≡ 1 must be satisfied.In fact, each step of the regression operation breaks this property.Therefore, a re-initialization operation is required to correct it.To prevent the computational cost of frequent re-initializations.An optimization object, as described in (12), is designed to correct the distortion of the LS function In summary, the optimization problem can be expressed as ( 13) λ phy , λ BC , λ data and λ LS control the relative importance of loss terms.The proposed network is informed by physics through an imposed penalty loss consisting of PDE, boundary condition, and LS optimization.By honoring the physical laws, the admissible solution space is significantly narrowed for the LS-PhyNet, rendering a greatly reduced requirement for the amount of training data.This practice allows the LS-PhyNet model, apart from utilizing sparse simulation data as labels, to eliminate the need for any other labels, thereby assisting the model to achieve better accuracy, generalization ability.In summary, in the SRM burning surface regression prediction task, the physics-constrained, boundary condition, and LS optimization losses play equally important roles of encouraging the prediction of burning surface state.These three losses are regarded as major losses.In contrast, for LS-PhyNet with small samples, the percentage of data loss is relatively small.Taking this into consideration, we set λ phy = λ BC = λ LS = 1, and λ data = 0.5 in our experiments.After obtaining the parameters of the network through penalized the optimization objective, a physics-guided LS-PhyNet model is built that not only fits the data, but also adheres to physical laws.

Calculation of the burning surface under uniform burning rates
The performance test of LS-PhyNet was conducted under uniform burning rates.The time steps and burning rates differed from those used in the training stage.400 iterations were performed to ensure that the SRM grain burned out.Figures 7-9 show the results of burning surface regression for the four types of propellants.(F, ∆t) corresponds to (1.6, 0.15), (2.8, 0.10), and (3.2, 0.08), respectively.The six columns in the figures correspond to 35, 50, 100, 150, 200, and 245 iterations.With an increase in the burned-off propellant web thickness, the configurations of grains 1-3 change significantly.The diameter of the inner hole of the grains increases, and the propellant web becomes very thin until it completely burns out during later stages.
Further, to evaluate the prediction results in the test phase, the burning surface area curves were obtained by calculating the burning areas of the traditional LS, PINN with fully-connected network [16], PINN with autoencoder network, minimum distance function (MDF) [38], and LS-PhyNet methods.PINN with autoencoder network follows the idea presented in [30], wherein the commonly used fully-connected architecture of PINN is replaced with an autoencoder.This autoencoder directly represents the solution to the PDE.The MDF method is based on the law of parallel-layer combustion, which holds that when a grain burns to a specific web thickness e, the burning surface at this moment should be the collection of all spatial points with the shortest distance of e between the interior of the grain and the initial burning surface.
The burning surface area is calculated by extracting the burning surface facets from the LS function φ .The task of extracting the burning surface area from the distance field φ is performed by the marching cubes algorithm [39].Figure 10 shows the comparison of the burning area change curves for above five methods.The curve of the traditional LS method is the reference benchmark.The LS-PhyNet method shows the best overall agreement with LS method and achieves a significantly lower error, highlighting its significant advantage in the burning process simulation.Following closely is the MDF method, exhibiting relatively good agreement with the reference benchmark.The variant PINN method shows a slight improvement over the PINN, yet a disparity remains compared with LS-PhyNet.This comparison underscores the superiority of the LS-PhyNet method in simulating the burning surface regression, which can produce physically reasonable and consistent results with high accuracy.Meanwhile, it provides insightful insights and inspiration for further research in this field.

Calculation of the burning surface under non-uniform burning rates
A performance test of the proposed LS-PhyNet was conducted under a non-uniform burning rate with cylindrical grain 4, as shown in figure 6.Two different burning rates were set, 1.0 for the low burning rate and 2.0 for the high burning rate.The total number of iterations was 300, and the time-step was 0.1.As shown in figure 11, because of the different burning rates, the degrees of regression are also diverse, forming clear steps on the burning surface.Over time, the front and rear grains differ significantly.The rear grain is nearly burned out, and the burning surface rapidly decreases, whereas a large burning surface still exists in the front grain.

Calculation of the burning surface with debonding
Debonding is a common defect in propellant grains.If the grains undergo debonding, abnormal burning surfaces and internal ballistics occur during the SRM operation, affecting the working performance of the  SRM.Therefore, the proposed LS-PhyNet was applied to test a grain with a debonding defect, where the burning rate was 1.0, the time-step was 0.1, and the total number of iterations was 350.Testing involved constructing a propellant grain with debonding defect, a scenario not present in the training set. Figure 12 shows the regression process for the initial cylindrical propellant charge with debonding.As burning continued, the pit formed by debonding became more evident owing to the lack of symmetry caused by debonding on one side, and it quickly connected to the inner hole.The burning area curves in figure 13 illustrate the significant differences between the grain with debonding defects and the normal condition, severely affecting the internal ballistics performance and SRM safety.

Calculation of the burning surface with new grains
To further verify the accuracy of the model, we also tested the new initial SRM grains, which were distinct from the training stage.As shown in figure 14, the uniform burning rate regression of grains 5-6 was predicted.F = 1.5, and F = 1.5.Figure 15 shows the burning surface area curves of the traditional LS method    and the LS-PhyNet method.It can be observed that the proposed LS-PhyNet method closely matches the LS method, with small errors.The results demonstrate the good generalization of the proposed model.

X Sun et al
Figure 16.Fake grain models.These five models are all selected from the grain 2 3D simulation models.

Computation efficiency of LS-PhyNet
The parameter and floating point operations (FLOPs) of the LS-PhyNet were computed to quantitative the computational complexity.The parameter is 4.8 × 10 5 and the FLOPs is 960 × 10 9 .The LS-PhyNet has some extrapolation ability, which can be used to predict solutions for cases with different initial conditions and burning parameters.Therefore, once trained, there is no need to retrain the LS-PhyNet repeatedly while changing some parameters, which is another advantage of the LS-PhyNet and helps to save a lot of computational cost.
Meanwhile, time required for one iteration of the LS-PhyNet and traditional LS solution methods were also determined.Note that all models are compared on the same hardware to eliminate the difference introduced by hardware.The proposed LS-PhyNet required 0.5371 s to perform one forward evaluation for each case, whereas the traditional LS solution required 312.6994 s.These values were averaged using multiple calculations to indicate the time required for one iteration.The traditional method is expected to require much longer than that required for LS-PhyNet because the traditional numerical solution method requires frequent re-initialization and evolution operator calculations, resulting in higher computational costs.Admittedly, LS-PhyNet method inevitably increases the training cost compared to the LS numerical solver, but the huge inference cost improvement and excellent generalization ability by doing so outweighs the training overhead.In fact, a complete regression calculation often requires hundreds of iterations.The computational cost of the numerical solution is prohibitive for scientific applications of burning surface regression calculations involving a large number of iterations.
Therefore, the computational cost of forward evaluation with the LS-PhyNet is much cheaper than running a LS solver, so the efficiency of SRM burning surfaces regression calculation is significantly improved.Although existing numerical techniques for solving PDEs are already mature nowadays, the result in this part demonstrates the promise of LS-PhyNet on modeling and simulating of complex PDE systems about SRM burning surfaces regression.

Applied to X-ray2CTNet
The proposed LS-PhyNet model has shown good generalization capability with a small amount of training data.To further demonstrate the potential of LS-PhyNet in practical ignition test applications, the prediction results were applied to ultra-sparse CT imaging, enabling 3D grain reconstruction from only two views.
The five corresponding burning states of grain 2 were selected as the training objects, and the simulation projections and CT volumes were obtained from full views.X-ray2CTNet was trained using these datasets.In the test phase, we pour the fake grain models shown in figure 16.The 5 groups of projection form two views are acquired by an YXLON FF20 CT system (figure 17) and used as test sets.Figure 18 shows the reconstructed results of the middle slice, and root mean square error (RMSE), structure similarity index measure (SSIM), and peak signal-to-noise ratio (PSNR) were calculated for the 3D volumes.The RMSE was 0.0169, PSNR was 35.9109, and SSIM was 0.985 735.The visual effects and computational metrics showed satisfactory reconstruction results.
In addition, we also discuss the robustness of the application of integrating the LS-PhyNet model with ultra-sparse CT imaging to other different grain charge configurations.The initial grain 1, grain 3, and four corresponding burning states were selected as the training models, and the simulation 2D x-ray projections and 3D CT volumes were obtained from full views.X-ray2CTNet was trained using these datasets.In the test phase, currents and voltages distinct from the training set were used to obtain two views simulation projections.Figure 19 shows the reconstructed and differences results of the middle slice.The RMSE was

and discussion
In this study, we recognized that sample shortages of SRMs highlights the need for monitoring accuracy, especially given the high cost associated with the design and development of SRM systems.We propose a novel approach called LS-PhyNet to effectively approximate the SRM burning surface regression calculation to construct SRM samples.Specifically, we modified the traditional fully connected network architecture of PINN and introduced a dense ED network for time evolution to learn PDE evolution operators.In addition, we addressed the lengthy re-initialization and evolution operator calculation times commonly observed in LS numerical solvers with a new optimization objective that combines into a single composite loss function from label data, physical constraints, and boundary conditions.This makes the network more reasonable.The effectiveness of the proposed method was demonstrated via a series of detailed experiments.The proposed LS-PhyNet model has shown good generalization capability with a small amount of training data.The proposed method enables the optimization of a technique applied to rapid monitoring of SRM burning surface transients to assess the accuracy and performance of SRM, which are critical to several scientific and engineering applications.
The combination of the neural operator, PINN and traditional solvers presents a promising research area, but there remains a gap between the accuracies of the LS-PhyNet and highly specialized numerical methods.Promising future directions include novel and effective optimization objectives, learning paradigms, and neural architectures to achieve more precise time-evolution modeling.We hope to assist or even replace traditional numerical PDE solvers by enabling faster high-fidelity simulations, optimizations, designs, and online control in numerous scientific and engineering applications.
In addition, our work tests the static regression case.If the proposed method is desired to be applied to dynamic monitoring scenarios, the model is required to have a good performance on the test set.The excellent generalization ability of the X-ray2CTNet is based on the assumption that the data distribution of the training set is consistent with that of the test set.Our work is also dedicated to this goal, but it was still done in a distributed manner.Regarding the actual testing of the SRM burning surface regression, future research should focus on integrating burning regression simulations with ultra-sparse CT imaging to achieve a one-stop solution.In other words, the regression simulation and reconstruction model are integrated into one-step training.Furthermore, the burning surface in propellant is influenced by various factors during the SRM operation, leading to erosion combustion effects.These complex issues will be considered in the subsequent work.

Figure 3 .
Figure 3. Dense block.The feature size remains unchanged and the channels increases or decreases.

Figure 4 .
Figure 4. Encoding-decoding block.Both the encoding and decoding blocks maintain the number of channels.

Figure 5 .
Figure 5. Encoding-decoding fusion (ED fusion) module.The low-level features correspond to the spatial attention part and the high-level features correspond to the channel attention part.

Figure 10 .
Figure 10.Comparison of the burning surface areas for grains 1-4.The horizontal coordinate is the burned-off web thickness and the vertical coordinate is the burning surface area.

Figure 11 .
Figure 11.Grain 4 burning process under a non-uniform burning rate.

Figure 12 .
Figure 12.Burning process of grain with debonding defect.Burning rate F = 1.0 and time-step ∆t = 0.10.

Figure 13 .
Figure 13.Comparison curve of the burning surface area with or without debonding defect.

Figure 15 .
Figure 15.Comparison of the burning surface areas for grains 5-6.The horizontal coordinate is the burned-off web thickness and the vertical coordinate is the burning surface area.

Figure 18 .
Figure 18.Slices of the 3D CT reconstruction results obtained from grain 2. The third row represents the differences between the above images.

Figure 19 .
Figure 19.Slices of the 3D CT reconstruction results obtained from grain 1 and grain 3. The second and fourth rows represent the differences between the ground truth and reconstruction images.